I think you’re on the right track, and @Carlos has already asked some great questions. I see the rod at top left, which looks like it connects to a damper, yes? The bricks look like firebrick, and I agree that if they are not, you need to replace at least the ones on the floor. My recollection is that the back of the original fireplace was already lined.
You are really going to have to soak that structure with heat. I’m still not clear from your description where you placed the ceramic fiber insulation. The stones for the fireplace will definitely warm over time, but the original fireplace design was not trying to refract and reflect heat back into the main burning space. So, if there is no thermal break between firebox and stones, they will still act as an enormous heat sink.
The opening is still enormous by oven standards. Would you consider closing off about 30 percent of that area (sides and top)? You could do that with firebrick (take a peek at the Mattone Barile directions, near the end where an optional closing build is described…it’s for an arch but it would be easier in your case).
And I agree, unless there is an insulative layer between the oven floor and the space underneath, wood or any other combustible material should not be stored there.
Keep talking to us, Dan! You’ve already got a 500-degree head start on this.